evolution - What if $rB=C$ in Hamilton's rule?

Hamilton's rule states that if $rB>C$ then a gene giving altruistic behaviour will increase in frequency in the population. What would happen if $rB=C$? Will an individual perform the altruistic act?

Answer

I agree with @Amory in the sense that Hamilton's rule is not a rule that applies to each specific individual and explain their behavior (or other traits). The Hamilton's rule describe the direction (and not the dynamic) of how a social traits evolve. A social trait is any trait which does not only affect the fitness of its carrier but also affect the fitness of other individuals in the population. Theoretically, all (or almost) traits are actually somehow social traits.

$RB>C$ is a simplistic way of looking at Hamilton's rule which might bring you to some confusion. This formula makes more sense to me by replacing $B$ by $\frac{dw(x,y,z)}{dx}$ and $C$ by $\frac{dw(x,y,z)}{dy}$. $w(x,y,z)$ is the fitness function of a focal individual expressing trait $x$ and interacting with an individual drawn from a subpopulation expressing trait $y$ (expected value of the probability distribution of trait expression of individuals in the subpopulation) in a population expressing trait $z$ (expected value again). In the particular case where there is no population structure $y$ equals $z$. $\frac{dw(...)}{dx}$ describes the partial derivative of the fitness function according to the variable $x$.

Therefore the Hamilton's rule can be rewritten as:

$$R\cdot\frac{dw(x,y,z)}{dx}>\frac{dw(x,y,z)}{dy}$$ where $R$ is the coefficient of relatedness which can itself be expressed as a correlation between the variables $x$ and $y$.

Altruistic behavior evolves if (but not "if and only if") this rule is respected. Hope this reformulation of Hamilton's rule helps a bit understanding it.

In the special case where $R\cdot\frac{dw(x,y,z)}{dx}=\frac{dw(x,y,z)}{dy}$ then, the trait of interest is not selected (neither counter selected). Therefore the trait will evolve under genetic drift alone. While the allele frequencies are modified by drift, the Hamilton's rule might (or might not) differ from the equality and then the trait would not be neutral anymore.